When someone has a fixed idea, it means that they are absolutely convinced of a particular viewpoint or belief, often to the point where it can be very difficult to persuade them otherwise. There are a number of synonyms that can be used to describe this kind of entrenched thinking, depending on the context. Some possibilities include obsession, preoccupation, fixation, single-mindedness, inflexibility, dogmatism, narrow-mindedness, and stubbornness. Other related phrases might include closed-mindedness, rigidity, or tunnel vision. Whether you are trying to describe a person's personality or a particular point of view, these synonyms can help to convey the idea of a fixed, unchanging perspective.